Theorem grpsubfvalg 13117

Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.)

Hypotheses

Ref	Expression
grpsubval.b	|- B = ( Base ` G )
grpsubval.p	|- .+ = ( +g ` G )
grpsubval.i	|- I = ( invg ` G )
grpsubval.m	|- .- = ( -g ` G )

Assertion

Ref	Expression
grpsubfvalg	|- ( G e. V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )

Distinct variable groups:   x, y, B   x, G, y   x, I, y   x, .+ , y

Allowed substitution hints:   .- ( x, y)   V( x, y)

Proof of Theorem grpsubfvalg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpsubval.m	. 2 |- .- = ( -g ` G )
2		df-sbg 13077	. . 3 |- -g = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) )
3		fveq2 5554	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpsubval.b	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2244	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5554	. . . . . 6 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		grpsubval.p	. . . . . 6 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2244	. . . . 5 |- ( g = G -> ( +g ` g ) = .+ )
9		eqidd 2194	. . . . 5 |- ( g = G -> x = x )
10		fveq2 5554	. . . . . . 7 |- ( g = G -> ( invg ` g ) = ( invg ` G ) )
11		grpsubval.i	. . . . . . 7 |- I = ( invg ` G )
12	10, 11	eqtr4di 2244	. . . . . 6 |- ( g = G -> ( invg ` g ) = I )
13	12	fveq1d 5556	. . . . 5 |- ( g = G -> ( ( invg ` g ) ` y ) = ( I ` y ) )
14	8, 9, 13	oveq123d 5939	. . . 4 |- ( g = G -> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) = ( x .+ ( I ` y ) ) )
15	5, 5, 14	mpoeq123dv 5980	. . 3 |- ( g = G -> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
16		elex 2771	. . 3 |- ( G e. V -> G e. _V )
17		basfn 12676	. . . . . 6 |- Base Fn _V
18		funfvex 5571	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
19	18	funfni 5354	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
20	17, 16, 19	sylancr 414	. . . . 5 |- ( G e. V -> ( Base ` G ) e. _V )
21	4, 20	eqeltrid 2280	. . . 4 |- ( G e. V -> B e. _V )
22		mpoexga 6265	. . . 4 |- ( ( B e. _V /\ B e. _V ) -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V )
23	21, 21, 22	syl2anc 411	. . 3 |- ( G e. V -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V )
24	2, 15, 16, 23	fvmptd3 5651	. 2 |- ( G e. V -> ( -g ` G ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
25	1, 24	eqtrid 2238	1 |- ( G e. V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   e. cmpo 5920   Basecbs 12618   +g cplusg 12695   invgcminusg 13073   -gcsg 13074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-sbg 13077

This theorem is referenced by:  grpsubval  13118  grpsubf  13151  grpsubpropdg  13176  grpsubpropd2  13177